{"id":24590938,"url":"https://github.com/matthewcaseres/omscs-bayesian-statistics","last_synced_at":"2026-02-16T02:38:28.333Z","repository":{"id":130649015,"uuid":"401186811","full_name":"MatthewCaseres/OMSCS-Bayesian-Statistics","owner":"MatthewCaseres","description":"Bayesian Statistics for OMSCS","archived":false,"fork":false,"pushed_at":"2021-08-30T03:22:09.000Z","size":386,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-10-10T23:12:13.965Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":null,"has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"other","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/MatthewCaseres.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null,"notice":null,"maintainers":null,"copyright":null,"agents":null,"dco":null,"cla":null}},"created_at":"2021-08-30T01:57:12.000Z","updated_at":"2021-08-30T03:22:12.000Z","dependencies_parsed_at":null,"dependency_job_id":"85af59c6-fe7f-49fb-a3a0-545fbc239e0c","html_url":"https://github.com/MatthewCaseres/OMSCS-Bayesian-Statistics","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/MatthewCaseres/OMSCS-Bayesian-Statistics","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/MatthewCaseres%2FOMSCS-Bayesian-Statistics","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/MatthewCaseres%2FOMSCS-Bayesian-Statistics/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/MatthewCaseres%2FOMSCS-Bayesian-Statistics/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/MatthewCaseres%2FOMSCS-Bayesian-Statistics/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/MatthewCaseres","download_url":"https://codeload.github.com/MatthewCaseres/OMSCS-Bayesian-Statistics/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/MatthewCaseres%2FOMSCS-Bayesian-Statistics/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":286080680,"owners_count":29498755,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2026-02-16T02:07:14.481Z","status":"online","status_checked_at":"2026-02-16T02:03:22.852Z","response_time":115,"last_error":"SSL_connect returned=1 errno=0 peeraddr=140.82.121.5:443 state=error: unexpected eof while reading","robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2025-01-24T09:44:33.059Z","updated_at":"2026-02-16T02:38:28.328Z","avatar_url":"https://github.com/MatthewCaseres.png","language":null,"funding_links":[],"categories":[],"sub_categories":[],"readme":"# Bayesian Statistics\n\nISyE6420 by [Brani Vidakovic](https://www.isye.gatech.edu/users/brani-vidakovic) is licensed under a [Creative Commons Attribution-NonCommercial 4.0 International License](https://creativecommons.org/licenses/by-nc/4.0/).\n\n**To read the math, either** \n1. Install a [Chrome plugin](https://chrome.google.com/webstore/detail/mathjax-plugin-for-github/ioemnmodlmafdkllaclgeombjnmnbima) to read math on GitHub directly.\n2. Use the VSCode Markdown previewer (supports math as of [version 1.58](https://github.com/microsoft/vscode-docs/blob/vnext/release-notes/v1_58.md#math-formula-rendering-in-the-markdown-preview)).\n\n\u003c!-- START doctoc generated TOC please keep comment here to allow auto update --\u003e\n\u003c!-- DON'T EDIT THIS SECTION, INSTEAD RE-RUN doctoc TO UPDATE --\u003e\n\u003ch2\u003eTable of Contents\u003c/h2\u003e\n\n- [Background](#background)\n  - [Classical vs. Bayesian Statistics](#classical-vs-bayesian-statistics)\n  - [FDA Guidance](#fda-guidance)\n\n\u003c!-- END doctoc generated TOC please keep comment here to allow auto update --\u003e\n\n# Background\n\nProfessor Vidakovic gives details on the life and work of Reverand Thomas Bayes. I am not sure we will be tested on any of this, but the curious student can find details on [wikipedia](https://en.wikipedia.org/wiki/Thomas_Bayes).\n\nBayes' relevance to Bayes Theorem is that one of his papers that was published posthumously has a special case of Bayes Theorem. Pierre-Simon Laplace generalizes this result to what we now recognize as Bayes' Theorem, (more on this later, don't worry)\n\n$$Pr(A_i | B) = Pr(A_i)\\frac{\\text{Pr}(B|A_i)}{\\sum_j \\text{Pr}(B|A_j)}$$\n\n\u003cdiv align=\"center\"\u003e\u003cimg style=\"background: white;\" src=\"svg/v8zRlR6V1G.svg\"\u003e\u003c/div\u003e\n\n## Classical vs. Bayesian Statistics\n\n![](./images/classical-vs-bayesian.png)\n\nSuppose we flip a coin 10 times and get 10 tails.\n\nIf the probability of getting heads is $p$ then the probability of this happening is $(1-p)^{10}$. The probability of this happening is maximized when $p=0$, and the frequentist approach will estimate that $p=0$.\n\nIn a Bayesian approach we find the probability distribution of our unknown parameter given some prior distribution. **We omit some details for now**.\n\nWe have an initial estimate of the probability distribution of $p$ as being uniform on the interval $[0,1]$. We update this estimate with the likelihood from our experiment and get a new density function of\n\n$$(1-p)^{10}*1$$\n\nWe normalize this density function to integrate to 1,\n\n$$11(1-p)^{10}$$\n\nWe can now get more meaningful information about the probability distribution like the\n\n- mean = 1/12\n- median = $(-.5)^{\\frac{1}{11}}+1$ = .061069\n- mode = 0\n\n## FDA Guidance\n\nProfessor Vidakovic summarizes [FDA guidance on the use of Bayesian statistics](https://www.fda.gov/media/71512/download).\n\nThe FDA is advocating for the use of Bayesian methods in development of medical devices.\n\n- Prior information from other devices with similar mechanisms of action is available and can be used by Bayesian methods.\n- Bayesian approaches may be simpler and less burdensome.\n- Prior information may reduce the required size of the trial.\n- Size of a trial can be reduced by early stopping\n  - Early stopping is more controversial when using frequentist methods compared to Bayesian\n- Multiplicity problems are easier when using Bayesian approaches.\n- Missing data is handled more gracefully by Bayesian approaches.\n- Unlimited looks at the data are okay in Bayesian approaches, more complicated in frequentist approaches and involve \"Alpha spending\"\n\nThe ability to stop the trial early due to **inferiority** (the method is not good) or **superiority** (the method shows itself to be good) reduces the human and economic costs of medical trials.\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmatthewcaseres%2Fomscs-bayesian-statistics","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fmatthewcaseres%2Fomscs-bayesian-statistics","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmatthewcaseres%2Fomscs-bayesian-statistics/lists"}